Their approach used the initial-state information by computing the Trotter error within the \(\eta \) -electron manifold, but the commutativity of the Hamiltonian was ignored, giving a simulation with gate count \(\left(n^{5/3}\eta ^{1/3}+n^{4/3}\eta ^{5/3}\right)n^{o(1)}\)  worse than our result. On the other hand, the work [1]} used commutativity of the Hamiltonian to show
\(\left\Vert {S}_p(t)-e^{-itH}\right\Vert _{\eta }=\mathcal {O}\left(\left(\left\Vert \tau \right\Vert _{\max }+\left\Vert \nu \right\Vert _{\max }\right)^{p-1}\left\Vert \tau \right\Vert _{\max }\left\Vert \nu \right\Vert _{\max }n^{p+2} t^{p+1}\right).\) 
